1. Biostatistics Lecture 7 4/7/2015

2. Chapter 7 Theoretical Probability Distributions

3. Outline 7.1 Probability Distribution7.1 Probability Distribution 7.2 The Binomial Distribution7.2 The Binomial Distribution 7.3 The Poisson Distribution7.3 The Poisson Distribution 7.4 The Normal Distribution7.4 The Normal Distribution 7.5 Z-score and Applications7.5 Z-score and Applications

4. 7.1 Probability Distribution

5. Random Variable variableAny characteristic that can be measured or categorized is called a variable. different values by chance random variableIf a variable can assume different values such that any particular outcome is determine by chance, it is called a random variable. probability distributionA probability distribution applies the theory of probability to describe the random variable.

6. Discrete and Continuous Random Variables discrete countableA random variable is discrete if it can assume a countable number of values. For example, the “coin” example assumes only 2 values – 1 and 0. continuousA random variable is continuous if it can assume an uncountable number of values. For example, a height or a weight, which can take on any value within a specified interval or continuum.

7. Probability Distribution probability distributionIn probability theory and statistics, a probability distribution identifies either each value –the probability of each value of an unidentified random variable (when the variable is discrete), or the value falling within a particular interval –the probability of the value falling within a particular interval (when the variable is continuous). Every random variable has a corresponding probability distribution.

8. Example P(X=4)=0.058 P(X=1 or X=2)=P(X=1)+P(X=2)=0.746 birth order of children born to women A discrete probability distribution of the birth order of children born to women in US (based on the experience of the US population in 1986). = Additive rule of probability for mutually exclusive events.

9. Comments In previous example, it is possible to tabulate the distribution because of limited count for this random variable. If a random variable can take on a large number of values, a probability distribution may not be a useful way to summarize its behavior. In this case, a number of summarization can help – population mean, population variance and population standard deviation.

10. Population Mean (Expected Value 期望值 ) Given a discrete random variable X with values x i, that occur with probabilities p(x i ), the population mean of X is For the case of rolling a dice, for example, we have

11. Population Variance Let X be a discrete random variable with possible values x i that occur with probabilities p(x i ), and let E(X) = μ. The variance of X is defined by

12. For the dice-rolling example

13. A brief summary This example tells you that, if you roll the dice many times, the average you may get is 3.5 points. It is likely that the average may ‘mostly’ be within the range 3.5±1.7 points.

14. pmf pdf pmf and pdf mass a discrete random variableIn probability theory, a probability mass function (abbreviated pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value. The graph of a probability mass function. All the values of this function must be non-negative and sum up to 1.

15. Cont ’ d probability density function continuousA pmf differs from a probability density function (abbreviated pdf) in that the values of a pdf are defined only for continuous random variables. integralIt is the integral of a pdf over a range of possible values that gives the probability of the random variable falling within that range.

16. a 1 =? a 2 =? a 3 =? a 4 =? Given a normal distribution a 2 =? a 4 =? 1 1 : area within IQR 2 2 : area between Q3 (or 0.6745  ) to Q3+1.5*IQR (or 2.698  ) 3 3 : area within  1  4 4 : area between 1  to 3 

17. Summary finite empirical probabilityProbabilities calculated based on a finite amount of data (such as the birth order example mentioned previously) are called empirical probability. The probability distributions for many other random variables of interest, however, can be determined (or approximated) based on theoretical consideration. theoretical probability distributionsThese are called theoretical probability distributions.